It's about as far from Vienna's Musikverein or the Wigmore Hall as you can imagine, but for the Takacs Quartet, it's home. This is Boulder, Colorado - defined on one side by the foothills of the Rocky Mountains, and with a pancake-flat plain that stretches right across the American Midwest on the other. It's sunnier here, they say, than in Miami, and in late October that's easy to believe; two days ago there was a snowstorm, but now the deepest of blue skies stretches overhead, and uncannily healthy-looking students wander through the honey-bricked university campus in shorts.

Yes, says cellist Andras Fejer, they could move to New York or London - but why would they want to? Any time saved on travel to concerts would be spent on day-to-day commuting. And besides, he wouldn't find it so easy to pop over to the tennis courts and work on his backhand.

It's refreshing to find a top-rank ensemble who are so focused on living as well as playing, and it certainly doesn't seem to be doing their career any harm. Reviewing their new Schubert CD in the Guardian a few weeks ago, Andrew Clements called the group "the greatest string quartet in the world". The praise is felt more keenly because the last couple of seasons have been ones of relative upheaval. The quartet now have a new record label in Hyperion, a new London performance base - they have succeeded the Alban Berg Quartet as Associate Artists of the South Bank Centre, where they return next week - and a new member in American viola player Geraldine Walther.

But how does a Hungarian string quartet end up in Colorado? The story has always been that the first lineup, students at Budapest's Franz Liszt Academy, met on a football pitch. Almost true, says Fejer, although three of them were already playing trios together before they found second violinist Karoly Schranz at that sports ground, and the Takacs Quartet (pronounced TOK-atsch) - named after their original first violinist, Gabor Takacs-Nagy - were born. "Karoly's soccer credentials were impeccable," explains a deadpan Fejer, while Schranz shakes his head and giggles; after 30 years, these two can still make each other laugh like schoolboys.

Success in international competitions came early; yet while the quartet were in a privileged position, able to travel abroad and to earn money in dollars, the payback came with the soul-destroying bureaucracy, the long hours spent waiting for passports rather than focusing on the music.

So when in 1982 a colleague asked if they could recommend a young string quartet for a residency at the University of Colorado in Boulder, the Takacs leapt at the opportunity. The Hungarian authorities weren't exactly happy, says Fejer, but decided to label the quartet as cultural ambassadors rather than defectors, smoothing the passage for the players and their families to move to the US.

Has the quartet's identity ever really hinged on their nationality? Fejer and Schranz don't consider it so, and while they enjoyed taking part in a Carnegie Hall recital celebrating the 50th anniversary of the Hungarian Uprising last month, they tend to play down this aspect. "Of course we adored playing Bartok," says Fejer, "but our education was cosmopolitan. Our heroes were the Amadeus Quartet."

These days, the Takacs are only half-Hungarian. When Takacs-Nagy left in 1993, the replacement the remaining quartet members chose was Edward Dusinberre, an English violinist fresh from the Juilliard School who didn't know a note of Bartok, but did come with a recommendation from the late, legendary teacher Dorothy Delay. Dusinberre was able to provide the leadership the others had realised they needed, and remains the group's natural spokesman. Meanwhile, Walther, who arrived from the San Francisco Symphony Orchestra, where she had been an immensely popular viola principal for nearly 30 years, is credited with bringing a new mellowness to the group.

Walther joined just after the quartet hit a peak with the completion of their award-winning Beethoven CD cycle, a defining challenge for any quartet. It might seem strange that the switch in record companies, from Decca to the independent Hyperion, came immediately afterwards - and, indeed, the quartet find it a tricky subject to discuss, since personnel changes at Decca, where they were signed for many years, have brought old friends back again. But when you consider that the Takacs had to partly finance the Beethoven recordings themselves, and that Decca had scheduled no future projects for them, it gets easier to understand. Hyperion director Simon Perry, meanwhile, says he is happy to give the Takacs carte blanche because "anything they record will be first-rate". Two Brahms CDs are scheduled over the next couple of years, including the Piano Quintet with Steven Hough. But Beethoven is still very much in their repertoire, and the Op. 132 Quartet will form part of their South Bank programme. "I'm enjoying playing it now more than when we recorded it," says Dusinberre. "During the recording there was a lot of talking and analysis going on, and at the time it had a good result. But language is unreliable where music is concerned. Now we're spending more time just listening to each other."

And what do they hear? "I think some quartets concentrate more on clarity and articulation," says Dusinberre, which is interesting considering how much the crispness of his own playing, bouncing off the warmer sounds from Schranz and Walther and the smoother buoyancy of Fejer's cello, define the quartet's sound. "We're probably more interested in the character. For us it's really important that there are four individual personalities communicating with each other - like actors on a stage."

Their current Schubert disc includes one of the most eloquent yet fierce readings of his Death and the Maiden you are likely to find: what made it that way? "I was kind of surprised by the ferocity of our first movement when I listened to it the first time," says Dusinberre. "We hadn't played the piece for about seven years. Maybe the experience of recording all the Beethovens made us want to pull the Schubert together structurally a bit more - but for whatever reason, we were tapping into its more desperate side. Schubert knew he was dying when he wrote it, so we thought more thoroughly about what it would mean to have that knowledge - the fear, the desperation, the desire for escape. We don't want people to be feeling too comfortable with it."

At the South Bank, the Takacs are emphatically not taking over from the Alban Berg Quartet, who are staying on as Quartet Laureate. But they will be assuming their schedule of three concerts a year, and will be performing South Bank commissions, including a new piece by James MacMillan. More excitingly, the quartet are also having a work written for them by Wolfgang Rihm, to be premiered in 2009.

Yet the quartet try to limit themselves to one new project a season, leaving them free to return again and again to the masterpieces of the repertoire. This, says Walther, is why she was so keen to join this ensemble: "They don't shy away from playing the core rep. I felt so many quartets didn't use that as their bread and butter, but it's the heart of our work."

Fejer is even more emphatic. "Musicologists tell us that composers knew their biggest thought processes would have to go into the string quartet. So can you imagine digging year after year into pieces by absolute geniuses who went the extra mile to produce something even better by their own extreme standards? It's bliss."

· The Takacs Quartet play at the Queen Elizabeth Hall, London SE1, on Thursday. Box office: 08703 800400. Schubert's Death and the Maiden is out now on Hyperion.